# Uncertainties and evaluations

## In a nutshell

Uncertainties in measurements can be calculated and are included in scientific reports. The evaluation of scientific investigations helps scientists to understand the significance of their research and plan further investigations.

### Equations

#### Uncertainty type | #### equation |

Analogue instrument | $uncertainty=\pm~\dfrac{smallest~ division ~on~ scale}{2}$ |

Digital instrument | $uncertainty =\pm~ smallest~division~on~the~scale$ |

Mean | $uncertainty =\pm~\dfrac{range~of~results}{ 2}$ |

## Uncertainties

Uncertainty is an estimated range of results between the measured value and the actual value. Uncertainty in measurements occurs due to random errors and limited resolution of instruments.

## Analogue instruments

The uncertainty of an analogue instrument can be calculated by using the following equation:

$uncertainty=\pm~ \dfrac{smallest~ division ~on~ scale}{ 2}$

##### Example

*A student uses a thermometer to measure the temperature of hot water. The smallest division on the scale is $0.05 \degree C$. Uncertainty of measurements using this thermometer:*

*$uncertainty = \pm(\dfrac{0.05}{2})=\underline {\pm~0.025\degree C}$*

## Digital instruments

The uncertainty of a digital instrument is simply the smallest division on the scale.

##### Example

*A student uses an electronic balance which reads to the nearest *$0.05~ g$*. The uncertainty of measurements using this balance is therefore $\underline{0.05~g}$.*

## Uncertainty of mean

The uncertainty of a mean result is calculated by the following equation:

$uncertainty =\pm~\dfrac{range~of~results}{2}$

### PROCEDURE

1. | Calculate the mean using the equation:
$mean=\dfrac{sum~of ~all~values}{ total ~number ~of ~values}$ |

2. | Calculate the range using the equation: $range=largest ~value - smallest~value$ |

3. | Calculate the uncertainty:
$uncertainty =\pm~(\dfrac{range~of~results}{ 2})$ |

4. | Calculate the uncertainty with the mean: $mean~\pm~ uncertainty$ |

##### Example

*A student measures the temperature of hot water. She repeats the experiment and records her results in a table. Work out the uncertainty with the mean. *

*repeat* | *temperature (*$\degree C$* )* |

$1$
| $38.5$ |

$2$
| $41.5$ |

$3$
| $39.0$ |

*Calculate the mean:*

$mean = \dfrac{(38.5+41.5+39.0)}{3}= 39.7\degree C$

*Calculate the range:*

$range=41.5-38.5= 3\degree C$

*Calculate the uncertainty to get the answer:*

$uncertainty~of~mean = \pm(\dfrac{3}{2}) = \underline{\pm~1.5\degree C }$

*The mean result is written with the uncertainty:*

$\underline{39.7~\pm1.5\degree C}$

## Evaluations

Results are evaluated in the discussion and conclusion of a report. The purpose of an evaluation is to work out:

- How well the experiment was designed

- How well the experiment was carried out
- The significance of scientific results
- How the results fit in with / contradict existing scientific literature

- Suggestions for future experiments

An evaluation includes consideration of:

- The repeatability of the investigation

- The reproducibility of the investigation
- The accuracy of results
- The precision of results

- Statistical tests

Suggestions for future experiments may include:

- Amendments for the experimental design
- Predictions for prospective results
- Advice for other scientists carrying out the same/similar experiments